# EVALUATE THE DEFINITE INTEGRAL 2 ∫dx/(x^2(√4x^2 + 9)) 1 INTEGRATION, SINGLE VARIABLE CALCULUS,TRIGONOMETRIC SUBSTITUTION

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You should factor out 9 under the square root such that:

int_1^2 (dx)/(x^2sqrt((4/9)x^2+1))

You should come up with the substitution x = (3/2)(cot x) =gt x^2 = (9/4)cot^2 x.

Differentiatingx = (3/2)(cot x)  with respect to x yields:

dx = -(3/2)(1/(sin^2 x))

int_1^2 (dx)/(x^2sqrt((4/9)x^2+1)) = int_1^2 (-(3/2)(1/sin^2 x)dx)/(((9/4)cot^2 x)sqrt(cot^2 x+1))

You ned to substitute 1/(sin^2 x)  for (cot^2 x+1)  such that:

int_1^2 (-(3/2)(1/sin^2 x)dx)/(((9/4)cot^2 x)sqrt(cot^2 x+1)) =int_1^2 (-(3/2)(1/sin^2 x))/(((9/4)cot^2 x)sqrt(1/(sin^2 x)))

int_1^2 (-(1/sin^2 x))/(((3/2)cot^2 x)(1/(sin x)))

int_1^2 (-(1/sin x)dx)/((3/2)cot^2 x)

You need to remember that cot x = cos x/sin x

int_1^2 (-(1/sin x))/((3/2)(cos^2 x)/(sin^2 x))

int_1^2 (-dx)/((3/2)(cos^2 x)/(sin x))

`(2/3)int_1^2...

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